# Fixed Ratio Position Sizing

Just a note about the fixed ratio position sizing method shared by Ryan Jones in his book The Trading Game. This mainly applies to trading futures.

Essentially, you add one contract once EACH of the existing contracts have brought in a certain amount of profit. The amount of profit required is called “delta”. This method ignores what the starting capital amount is.

Example:

• Delta = \$5,000, you are currently trading 2 contracts, and you have a \$40,000 capital base.
• To be able to trade 3 contracts, each of the 2 contracts must make \$5,000 of profits, for a total of \$10,000 profits, before you are allowed to add the 3rd contract to your trading size.
• 4th contract will  be added when your capital reaches \$50,000 + 3*\$5,000 = \$65,000
• 5th contract will  be added when your capital reaches \$65,000 + 4*\$5,000 = \$85,000
• And so on…
• However once your capital drops back below \$85,000, you need to reduce your contract size back to 4. And when your capital drops below \$65,000, you need to reduce your contract size back to 3.

There is a mathematical formula to calculate the number of contracts that you can trade.

• If you started off with capital K and traded 1 contract, and let the delta be D. Then
• First calculate P = Current capital – K. That is your accumulated profits from your original capital.
• Number of contracts to trade N = 0.5 + 0.5 * Sqrt ( 1 + 8 * P / D )
• If you started off with capital K and traded X contracts, and delta is D. Then
• P = Current capital – K
• Number of contracts to trade N = 0.5 + 0.5 * Sqrt ( 4*(X^2) – 4*X + 1 + 8 * P/D )

For those who are curious, the derivation of the formulas above goes like this

• Assume original capital = K0, original number of contracts traded = L0
• K1 = K0 + L0 * D
• K2 = K1 + L1 * D = K0 + D * (L0 + L1)
• K3 = K2 + L2 * D = K0 + D * (L0 + L1 + L2)
• K3 – K0 = D * (L0 + L1 + L2)
• P = D * (L0 + L1 + L2)
• P / D = Sum of arithmetic progression starting with L0 = 0.5 * n * ( 2*L0 + (n – 1) ) where
• n is the number of terms
• L(n) = L(n-1) + 1.
• Now re-arrange the terms, solve the quadratic equation, and you will get the formulas above.

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